#398349

order	512
nilpotency class	3
isoclinism	isoclinic class of #394358
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60425	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1592 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

(2,3)(5,6)(7,11)(8,19)(9,14)(10,13)(12,21)(15,23)(16,24)(17,25)(18,26)(20,32)(22,29)(27,31)(28,34)(30,35)(33,36);(1,2,4,3)(5,7)(6,11)(8,20)(9,18)(10,17)(12,24)(13,25)(14,26)(15,27)(16,21)(19,32)(22,33)(23,31)(28,35)(29,36)(30,34);(5,8)(6,12)(7,16,18,20)(9,21)(10,22)(11,24,26,32)(13,28)(14,19)(15,29)(17,30,31,33)(23,34)(25,35,27,36);(1,3,4,2)(6,13)(10,23)(11,25)(12,28)(14,15)(17,31)(19,29)(22,34)(24,35)(26,27)(30,33)(32,36)

Polycyclic presentation:

2^9|b^2=e,c^2=fi,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,f^b=fi,g^c=gi,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 305545);